View
224
Download
1
Category
Preview:
Citation preview
1
2.4 Linear Independence (線性獨立 ) and Linear Dependence(線性相依 )
A linear combination( 線性組合 ) of the vectors in V is any vector of the form c1v1 + c2v2 + … + ckvk where c1, c2, …, ck are arbitrary scalars.
A set of V of m-dimensional vectors is linearly independent( 線性獨立 ) if the only linear combination of vectors in V that equals 0 is the trivial linear combination.
A set of V of m-dimensional vectors is linearly dependent ( 線性相依 ) if there is a nontrivial linear combination of vectors in V that adds up to 0.
(p.32)
2
Example 10: LD Set of Vectors (p.33)
Show that V = {[ 1 , 2 ] , [ 2 , 4 ]} is a linearly dependent set of vectors.
Solution Since 2([ 1 , 2 ]) – 1([ 2 , 4 ]) = (0 0), there
is a nontrivial linear combination with c1 =2 and c2 = -1 that yields 0. Thus V is a linear dependent set of vectors.
3
What does it mean for a set of vectors to linearly dependent?
A set of vectors is linearly dependent only if some vector in V can be written as a nontrivial linear combination of other vectors in V.
If a set of vectors in V are linearly dependent, the vectors in V are, in some way, NOT all “different” vectors. By “different” we mean that the direction specified by
any vector in V cannot be expressed by adding together multiples of other vectors in V. For example, in two dimensions, two linearly dependent vectors lie on the same line.
Linear dependent 線性相依 (p.33)
4
Let A be any m x n matrix, and denote the rows of A by r1, r2, …, rm. Define R = {r1, r2, …, rm}.
The rank( 秩 ) of A is the number of vectors in the largest linearly independent subset of R.
To find the rank of matrix A, apply the Gauss-
Jordan method to matrix A. (Example 14, p.35) Let A’ be the final result. It can be shown that the
rank of A’ = rank of A. The rank of A’ = the number of nonzero rows in A’. Therefore, the rank A = rank A’ = number of nonzero rows in A’.
The Rank of a Matrix (p.34)
5
A method of determining whether a set of vectors V = {v1, v2, …, vm} is linearly dependent is to form a matrix A whose ith row is vi. (p.35)
- If the rank A = m, then V is a linearly independent set of vectors.
- If the rank A < m, then V is a linearly dependent set of vectors.
Whether a Set of Vectors Is Linear Independent
6
2.52.5 The Inverse of a Matrix ( 反矩陣 ) (p.36) A square matrix( 方陣 ) is any matrix that has
an equal number of rows and columns. The diagonal elements ( 對角元素 )of a
square matrix are those elements aij such that i=j. A square matrix for which all diagonal elements are
equal to 1 and all non-diagonal elements are equal to 0 is called an identity matrix( 單位矩陣 ). An identity matrix is written as Im.
7
The Inverse of a Matrix (continued)
For any given m x m matrix A, the m x m matrix B is the inverse of A if BA=AB=Im.
Some square matrices do not have inverses. If there does exist an m x m matrix B that satisfies BA=AB=Im, then we write B= . (p.39)
The Gauss-Jordan Method for inverting an m x m Matrix A is Step 1 Write down the m x 2m matrix A|Im Step 2 Use EROs to transform A|Im into Im|B. This
will be possible only if rank A=m. If rank A<m, then A has no inverse. (p.40)
1A
8
(p.41)
Matrix inverses can be used to solve linear systems. (example, p.40)
The Excel command =MINVERSE makes it easy to invert a matrix. Enter the matrix into cells B1:D3 and select the
output range (B5:D7 was chosen) where you want A-1 computed.
In the upper left-hand corner of the output range (cell B5), enter the formula = MINVERSE(B1:D3)
Press Control-Shift-Enter and A-1 is computed in the output range
9
Inverse Matrix ( 反矩陣 )
10
反矩陣之性質
p.42 #9
p.42 #10
p.41 #8a
11
p.41 problems group 8a
12
Example :
例題:
13
Example (continued)
Exercise : problem 2, p.41, 5min
14
Orthogonal matrix (正規矩陣 )
A-1=AT
Det(A)=1 or Det(A)=-1
Determine matrix A is an orthogonal matrix or not .
cossin
sincosA
p.42 #11
15
AX=b 為一線性方程組 , 若 b1=b2=…=bm=0 ,則稱為齊次方程組(homogeneous) , 以 AX=0 表之。
1.x1=x2=…=xn=0 為其中一組解,稱為必然解 (trivial solution)
2.若 x1 、 x2 、… xn 不全為 0 ,則稱為非必然解 (nontrivial
solution)
Homogeneous System Equations
16
17
Example: ? solution trival a has
101
011
326
0 A,AX
solution. trivala has system the
00A]x,x,[xX
matrix,singular non isA 1.1-T
321
solution. trival a has system the ~
~~~ ~
~~~ 2
,]xxx[X
.
T 0
0100
0010
0001
0100
0010
0101
0100
0110
0101
0010
0110
0101
0320
0110
0101
0320
0011
0101
0326
0011
0101
0326
0011
0101
0101
0011
0326
321
18
Example: ?solution trivala has
222
012
111
,0
AAX
solution. nontrival a has system the 3
23
3
2
1
Rt
tx
tx
tx
19
20
Example : Solving linear system by inverse matrix
21
22
23
24
Exercise : Solving linear system by inverse matrix
1x x 3x
62xx 5 2x
7x x 2 x
321
321
321
19x 8,x 4,x 321 Answer:
(8 min)
25
2.6 Determinants(行列式 ) (p.42)
Associated with any square matrix A is a number called the determinant of A (often abbreviated as det A or |A|).
If A is an m x m matrix, then for any values of i and j, the ijth minor of A (written Aij) is the (m - 1) x (m - 1) submatrix of A obtained by deleting row i and column j of A.
Determinants can be used to invert square matrices and to solve linear equation systems.
26
Determinants(行列式 )
27
Minor( 子行列式 ) & Cofactor( 餘因子 )
28
餘因子展開式
29
行列式之性質 (1)
30
行列式之性質 (2)
31
行列式之性質 (3)
32
行列式之性質 (4)
33
Adjoint matrix ( 伴隨矩陣 )
34
Exercise :
(1) adj A = ?
(2) det(A) = ?
231
540
213
A
12104
1545
1387
Aadj det(A) = -34
Answer :
35
Exercise :
若 det(λI3-A)=0, λ= ?
212
030
313
A
λ =-5 或 λ=0 或 λ=3
36
伴隨矩陣之性質
37
行列式之性質
38
Cramer’s Rule
39
Cramer’s Rule 注意事項若 det(A) ≠0 ,則 n 元一次方程組為相容方程組,其唯一解為
若 det(A) =det(A1) =det(A2)= …=det(An) =0 ,則 n 元一次方程組為相依方程組,其有無限多組解
若 det(A) =0 ,而 det(A1) ≠0 或 det(A2) ≠0 ,…,或 det(An) ≠0 ,則 n 元一次方程組為矛盾方程組,其為無解。
)det(
)det( ..., ,
)det(
)det( ,
)det(
)det( 22
11 A
Ax
A
Ax
A
Ax n
n
40
Example : Solving Linear System
635
212
211
A
6635
12 2
42
321
321
321
xxx
xxx
xxx
Sol:
0
635
112
411
0
665
212
241
0
636
211
214
0
635
212
211
3
21
)Adet(
,)Adet(,)Adet(,)Adet(
The system has an infinite number of solutions
41
Exercise : Solving linear system by Cramer’s rule
1x x 3x
62xx 5 2x
7x x 2 x
321
321
321
19x 8,x 4,x 321 Answer:
42
LU-Decompositions
Factoring the coefficient matrix into a product of a lower and upper triangular matrices.
This method is well suited for computers and is the basis for many practical computer programs.
43
Solving linear systems by factoring
Let Ax = b , and coefficient matrix A can be factored into a product of n × n matrices as A = LU , where L is lower triangular and U is upper triangular, then the system Ax = b can be solved by as follows :
Step 1. Rewrite Ax = b as LUx = b (1)
Step 2. Define a n × 1 new matrix y by Ux = y (2)
Step 3. Use (2) to rewrite (1) as Ly = b and solve this system for y.
Step 4. Substitude y in(2) and solve for x
44
Example : Solving linear system by LU decomposition
3
2
2
100
310
131
734
013
002
100
310
131
734
013
002
294
083
262
3
2
1
x
x
x
3
2
2
294
083
262
3
2
1
x
x
x
Step 1. A=LU, Ax=b LUx=b
45
Step 2. Ux=y
Step 3. Ly=b
Solving y by forward-substitution. y1=1, y2=5, y3=2
37y3y 4y
2 y 3y
2 2y
3
2
2
734
013
002
321
21
1
3
2
1
y
y
y
3
2
1
3
2
1
100
310
131
y
y
y
x
x
x
46
Step 4.
y1=1, y2=5, y3=2
Solving y by backward-substitution. x1=2, x2=-1, x3=2
2x
53xx
1x 3xx
2
5
1
x
x
x
100
310
131
3
32
321
3
2
1
3
2
1
3
2
1
100
310
131
y
y
y
x
x
x
47
Assume that A has a Doolittle factorization A = LU.L : unit lower-△ , U : upper-△
=
The solution X to the linear system AX = b , is found in three steps: 1. Construct the matrices L and U, if possible. 2. Solve LY = b for Y using forward substitution. 3. Solve UX = Y for X using back substitution.
Doolittle method
=
48
=
Assume that A has a Doolittle factorization A = LU. L : lower-△ , U : unit upper-△
The solution X to the linear system AX = b , is found in three steps: 1. Construct the matrices L and U, if possible. 2. Solve LY = b for Y using forward substitution. 3. Solve UX = Y for X using back substitution.
Crout method
49
Solving Linear Equations (see *.doc) Gauss backward-substitution 高斯後代法 -- forward-substitution Gauss-Jordan Elimination 高斯約旦消去
法 -- reduced row echelon matrix Inversed matrix ─ solve Ax=b as x=A-1b LU method ─ solve Ax=b as LUx=b Cramer's Rule ─ x1=det(A1)/det(A),….
Recommended